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f = Ew,r,, where r, is the rate of return on stock n.
The set of weights for each strategy has the following characteristics: (1) The weights are both positive and negative, and
the sum of the weights is zero. Consequently, the return to the strategy can be viewed as the return on a ”pure hedge portfolio” with a zero invest- ment value. The weights are constructed so that the sum of the weights is zero within each of 55 industry groups. Each strategy therefore takes both long and short positions in each industry, which av- erage out to zero, and so is immunized against industry factors of return. The strategy is also constructed to be orthogonal to a set of ”risk indexes,” with which common factors of return are also associated. The weighted sum of each of the following risk indexes, weighted by the strategy weights, is zero: 1.
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Variability in Markets. Beta prediction based upon stock price behavior, option price, etc. Success. Past success of the company, as measured by stock’s performance and earn- ings growth. Size. A size index based on assets and capi- talization. Trading Activity. Indicators of share turn- over. Growth. A predictive index for subsequent earnings per share growth. EarningslPrice. Ratio of estimated current normal earnings per share to stock price. Earnings Variation. Variability of earnings and cash flow. Financial Leverage. Balance sheet and oper- ating leverage of industrial companies. Foreign Income. Proportion of income identified as foreign. Labor Intensity. Ratio of labor cost to capi- tal cost. Yield. Predicted common stock dividend yield.
Consequently, the return to the strategy is im- munized against any common factor returns as- sociated with these stock characteristics. The booklprice and specific-return-reversal strat- egies are orthogonal to one another. The two sets of weights have zero cross-product. Conse- quently, the return on each strategy is expected to be independent of the other one. Each strategy is standardized, so as to imply an exposure to the variable that is constant over time.
For the book/price strategy, the weighted sum of book/price ratios diflers from the market average by one cross-sectional standard deviation of that ratio. In other words, the strategy is persistently located one standard deviation away from the capitalization-weighted mean value for all stocks. For the specific-return-reversal strategy, the sum of the positive weights is 1.0, and the surn of the negative weights is -1.0, so the return on the strategy corresponds to the difference between returns on a ”buy psortfolio” of stocks with neg- ative prior specific r12turns and a ”sell portfolio” of stocks with posxtive prior specific returns. (With respect to an ”indicator variable” for the sign of the previous month’s specific return, this strategy is positioned at two cross-sectional stan- dard deviations away from the mean, SCI that it is, in a precise sense, twice as aggressive with respect to its instrurnental variable as the book/ price ratio strategy is with respect to its instru- mental variable.) The set of weights for each strategy is calculated so as to minimize the variance of the strategy’s return arising from the specific returns of the in- dividual companies, :subject to meeting the above five restrictions. In other words, the noise re- sulting from the random specific returns of the individual stocks is made as small as possible.
Because each strategy is a “pure hedge portfolio,” we can view the return to the strategy as a potential in- cremental return that an investor can earn by adjust- ing an existing portfolio in the direction of the strategy.
Let h, denote the investment proportions in an ordinary portfolio of common stocks. Let ro denote the investment rate of return on that portfolio. Then if the initial portfolio is adjusted in the direction of the hedged portfolio, so t:hat the resulting investment weights are each (h, + w,), then the rate of return on the adjusted portfolio will be ro + f. For this reason, statistically significant performance of the strategy - to the extent that that performance is un- correlated with the return on the initial portfolio - implies that it is necessarily possible to improve the meanhriance characteristics of the initial portfolio by making the adjustment, and so suggests that the investor holding portfolio weights h, would prefer to hold portfolio weights h, + w,; thus, good perform- ance suggests an inefficiency in the marketplace.
THE TWO STRATEGIIES AS INSTRUMENTS FOR MARKET INEFFICIENCY
Suppose that the rnarket is in fact inefficient, in the sense that if v, is the ”fair value” of stock n,
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then the stock price p, differs from the fair value by a pricing error en, i.e., pn = v, + e,. The usual pre- sumption is that the market price is unfair in the sense that the pricing error e will be reversed in the future. Consequently, the rate of return in the subsequent month, r,, is negatively correlated with e,. A variable, x,, will serve as an “instrumental variable” for sub- sequent performance, r,, if it is correlated with the initial pricing error, e,. Therefore, to search for market inefficiency, we should search for a variable, x, which we expect to be negatively correlated with e, and therefore positively correlated with subsequent re- turn, r. This variable will define the strategy that tests for the existence of the pricing error, by means of the test of subsequent returns.
One way to obtain an instrument for e is to find a variable that is correlated with the difference v - p, since -e = v - p. For a variable x to be pos- itively correlated with v - p, x must increase when the value of the firm increases relative to the price of the firm.
Traditionally, ratios of the firm’s activity to the stock price have been used for this purpose. In prin- ciple, any ratio, such as booklprice, earnings/price, or dividendlprice = yield, can be used. Nevertheless, the value of these financial ratios as instruments may be destroyed if they are used in the process of security analysis or as a quantitative screen by investors using quantitative techniques.
If an investor uses the variable x as an indi- cation of the desirable stock quality, so that stock price is bid up in proportion to x, then x may acquire a positive correlation with p, over and above the in- direct relationship with p, which x obtains through its link to underlying value, v. As the correlation with p increases (as the stocks with high x values are bid up in price and stocks with low x values are bid down in price), the result is to reduce the correlation of x with v - p and eventually to destroy its usefulness entirely. Since substantial work had previously been done with yield as a criterion for investment, and since the earningdprice ratio was much emphasized in security analysis and had previously been studied in the finance literature by S. Basu, we felt that the book/price ratio was an intriguing candidate for study. Since it had not been heavily described in the quantitative literature, it might possibly serve as an as-yet unspoiled instrument.
Another approach to obtaining an instrumental variable is to attempt to find a variable x that is directly correlated with the pricing error e. The previous months specific return, u,,~.~, is a natural instrument for this purpose.
The explanation of this relationship is straight-
forward. Suppose that a common-factor model is used to fit the most probable return for this stock in the previous month, by analogy with the returns with similar stocks. In other words, the common-factor model explains the returns on all stocks as a result of their characteristics, and so estimates factors of return associated with industry groups and with risk in- dexes. Then, to the extent that the stock’s previous month’s return differed from this fitted return, the difference was unique to that stock. If there is a pric- ing error for the stock, that error would probably show up as a component of this unique return.
In fact, we can consider the difference between the pricing error for the stock at the end of the prior month and the pricing error at the inception of that month as one of the components of the previous month‘s specific return. Therefore, in the absence of some adjustment to remove this relationship, we would expect that the previous month’s specific re- turn would be positively correlated with every one of its components and, particularly, with the component that was the change in the pricing error.
The final step in the argument is to notice that the pricing error at the end of the previous month is the starting point for the current month’s return: A larger change in pricing error over the previous month implies, ceteris paribus, a likelihood of a larger pricing error at the end of the previous month.
The complete linkage is as follows: The pre- vious month’s specific return is positively correlated with its component, which is the change in the pricing error over the previous month, which is positively correlated with the magnitude of the pricing error at the end of the previous month. Therefore, the pre- vious months specific return is intrinsically positively correlated with the pricing error at the end of the previous month. Consequently, we can expect the negative of the specific return to be positively corre- lated with this months investment return.
As in the case of the book/price variable, we must ask whether this correlation would be vitiated by use of the previous month‘s specific return by tech- nicians as a transaction strategy. In other words, if market participants were actively seeking to profit from anticipated specific return reversals, the results would be to reduce, and even eliminate, the use of the instrumental variable.
There are two reasons, however, to think that the instrument might remain valid. First of all, be- cause the strategy requires a high rate of turnover, the inhibition provided by transaction costs could leave a significant correlation even if the investment value of the strategy had been fully removed. Second, because of the strong bias toward market efficiency
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that has been present in academic circles, there might be skepticism about the use of such a simple, tech- nical, quantitative rule for trading strategies.
For these reasons, we felt that the booklprice (B/P) strategy and the specific-return-reversal (SRR) strategy were natural instruments to use in the search for market inefficiencies.
41 4.5 83
IMPLEMENTATION OF THE STRATEGIES ’
AND CALCULATION OF THE RESULTS
We based the initial retrospective te$t of these strategies on a data base of monthly stock data from January 1973 through March 1980 for the B/P strategy, and on through December 1980 for the SRR strategy. For the retrospective study, we strove to assure that all data used in calculation of the weights in the strat- egies would have been available prior to the month for which the return was calculated. We also carefully screened the data base to remove as many errors as possible, so that the investment returns would be
We based this analysis primarily upon the Standard & Poor’s Compustat data base and the IBES Analytics data base. There was no retrospective bias in the latter, and retrospective bias in the former could be avoided by use of the Compustat Research Tape. As a result, we were able to avoid survivorship bias and retrospective inclusion bias.
For present purposes, the key concern is with the prospective tests, beginning with the endpoints of the retrospective studies. Strategy weights for every month were calculated, based upon data through to the prior month’s close, and calculation of the strategy weights was usually completed by the second or third business day of the month. The sam- ple was defined prospectively as the HICAP universe. The strategic returns calculated here are therefore a true test of the outcome of a predefined investment strategy.
12 % valid. 5 - B 0
32 36 3.7 i5.7 62 76
PERFORMANCE OF THE BOOWPRICE STRATEGY
The monthly strategy returns f, can be analyzed for their relationships with the market returns by means of the time-series regression:
f, = a + PrMt + et, t = 1, . . ., T (1)
where rMt is the excess return on the market (the monthly S&P 500 return minus the monthly 30-day Treasury Bill return), and E, is the unexplained return. The coefficient p gives the responsiveness of the strat- egy return to the market portfolio, and a is the av- erage residual factor return. Let o denote the standard deviation of the residual return, w = std. dev. (E).
Table 1 summarizes the results of this regres-
Number of months positive 64 Number of months negative 23 Number of months total 87
TABLE 1
Monthly Performance of the BooWPrice Strategy
38 102 16 39 54 i.41
1973.1- 1980.3 1984.9 1984.9
a (basis points) t-statistic w (basis points) ’____
sion for the 87 months of the retrospective study, for the 54 months of the prospective study, and ,for the total sample of 141 months. Each panel provides the average residual return (IZ) for this strategy and the standard deviation of the residual return (w), in basis points per month. For example, the average residual return for the entire period was a = 36 basis points, or 0.36 percent per month, and the standard deviation of the monthly residual return was 76 basis points. The systematic risk coefficient, p, was indistingulsh- able from zero, so it is not reported in the table. The foot of Table 1 shows the number of monthly returns that were positive, negative, and the total for each subperiod and for the entire history.
The return to the EM’ strategy was positive in 38 of the 54 months of the prospective evaluation. The mean residual return was 32 basis points and the standard deviation of monthly residual return was 62
permits us to reject the hypothesis that the mean re- sidual return is zero at the 99.95% level of confidence. The performance of the B/P strategy in the evaluation period was consistent with the prior expesience. Therefore, we are justified in combining the entire sample history into a single test of market efficiency.
Table 2 shows an intriguing aspect of the B/P returns for the 12 calendar months. The left-hand
basis points. This led to a t-statistic of 3.7, which
TABLE 2
Seasonality of Book/Price Returns (Basis Points)
1973.1-1980.3
w u t-stat
January February March April
June May
July August September October November December
193 125 ( 4.39) 37 45 ( 2.31) 50 87 ( 1.63) 18 30 ( 1.63) 21 40 ( 1.40) 36 40 ( 2.43)
47 61 ( 2.05) 20 68 ( 0.78) 43 55 ( 2.07)
33 75 ( 1.16) -28 69 (-1.08)
-13 42 (-0.81)
- 1980.4-1984.9
p. u t-stat
133 62 ( 4.29) :77 42 ( 3.67) 47 67 ( 1.39) 47 40 ( 2.64) 23 34 ( 0.85)
-17 53 (-0.72)
39 39 ( 2.22) -13 86 (-0.33)
i o 75 ( 0.30)
38 44 ( 1.75) 25 29 ( 1.71)
-16 23 (-1.39)
-- 1973.1-1384.9
u t-stat
173 109 ( 5.58)
49 78 ( 2.18) 30 36 ( 2.88) 22 36 ( 2.15) 14 51 ( 0.97)
44 !jl ( 2.97) 6 74 ( 0.215)
29 63 ( 1.61) -24 55 ( - 1.45)
35 63 ( 1.85) 1 41 ( 0.05)
50 47 ( 3.70)
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panel shows the mean and standard deviation of the returns over the historical sample. Both the mean (p) and the standard deviation (a) of the boowprice re- turn were much higher in January than in any other month. There appears to be a downward trend in IJ. over the course of the year. As the monthly t-statistics in the left-hand panel show, the mean return was highly significant in January (t-statistic = 4.39), and the t-statistic exceeded 2 in February, June, July, and September. We emphasized this seasonal pattern in our discussions of the strategy in 1982 [ll].
The central panel of Table 2 displays the monthly means and standard deviations during the prospective evaluations. Again, the January mean stands out sharply and, again, there is an appearance of a downtrend in the mean values from January through December. Despite the brevity of the sample, the January and February means achieve high statis- tical significance, and the April and July means have t-statistics greater than 2.0.
The right-hand panel shows the seasonality for the entire eleven- and-three-quarter year sample. Here the downward trend from January through to the end of the year is pronounced, and the t-statistics for January, February, March, ApriI, May, and July are each separately greater than 2.0.
1973.5- 1980.12
PERFORMANCE OF THE SPECIFIC- RETURN-REVERSAL STRATEGY
The SRR strategy defined in the earlier paper [lo] (Rosenberg and Rudd (1982)) used the negative of the previous months specific return as the instru- mental variable. Table 3 reports the strategy reported in the earlier paper, together with the subsequent performance of the strategy,
1981.1- 1973.5- 1984.10 1984.10
Number of months positive 83
Number of months total 92 Number of months negative 9
p (basis points) t-statistic u (basis points)
43 126 3 12
46 138
112 104 109 10.4 I 10: 1 13.83 103 93
The performance in the prospective evaluation is similar to the historical study. The mean monthly return is smaller, but the time-series variability of the return is reduced even more, so that the strategy achieves even higher significance per unit time after the prospective evaluation. In fact, the results are pos-
itive 43 months out of 46. The result is a t-statistic of 10.3, which permits an essentially conclusive rejection of the null hypothesis that the actual mean return of the strategy is 0.0.
To provide a still clearer strategy, and to in- sulate the results from the effects of misrecorded prices, we considered an alternative strategy in which the instrumental variable is the sign of the previous month’s specific return. In other words, the strategy is simplified to purchasing an equal-weighted “buy portfolio” of stocks whose previous month’s specific returns were negative and selling short an equal- weighted portfolio whose previous months specific returns were positive. The monthly return on that strategy is simply the difference between the monthly returns for the buy and sell portfolios, which coin- cides with the difference between the average return for the month on the stocks whose previous months specific returns were negative and the average return in the month for the stocks whose previous month’s specific returns were positive. The results of that strat- egy appear in Table 4. As the beta was significantly different from zero, we carried out the time-series regression on the market return (Equation 1) and re- port the alpha, beta, and residual standard deviation, omega, in the table. This strategy achieves an even higher level of statistical significance, with a t-statistic of 11.5 for the 46-month sample. The results are pos- itive 45 months out of 46. Average January abnormal profits were 202 basis points, versus 129 basis points on average for the other eleven months of the year. This difference is intriguing, but it was not statistically
13 I-
i.5 3 9 3 E -1
r; 2 8 a 5 3 0,
significant. z TABLE 4
SRR Monthly Return (Basis Points)
a P w
136 0.10 80 (11.54) (3.65)
t-statistics in parentheses.
TRADING THE STRATEGIES
Trading costs are an important aspect to be considered in applying these strategies. Trading costs include the direct expenses of commissions and taxes, plus the price effect of trading. Trading costs for an institutional investor utilizing the B/P strategy would almost certainly have had a negligible effect upon performance. Urgent trading of the B/P strategy is not necessary, because the B/P criterion variable is not timely; a round-trip trading cost of 100 basis points is probably an ample allowance. Portfolio turnover is
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less than 5% per month, so that the drain from trading costs would be less than 5 basis points per month, as against an average abnormal performance of 36 basis points per month for the entire history.
The performance of the SRR strategy, on the other hand, would be greatly reduced for an investor experiencing trading costs. The strategy relies on timely data, so that urgent trading is important. Since the SRR strategy reported in Table 4 involves holding one portfolio long and another portfolio short, and since approximately 50% of the stocks in each port- folio are switched each month, there is a trading cost drain equal to 100% of the round-trip trading cost each month. Therefore, a drain of 100 basis points or more against a monthly performance of 136 basis points is not unlikely.
Some investors would not be faced with these trading costs. Brokers and dealers, for example, might face trading costs that were a fraction of this. Also, the investor who had determined to trade for other reasons, and who was using the SRR strategy as a timing device, would face no incremental trading
The abnormal return of 136 basis points per month reported in Table 4 for the SSR strategy may be unobtainable if an investor is unable to sell short the “sell portfolio” at the month-end closing prices.’ We evaluated an alternate strategy where the investor takes a long position in the “buy portfolio” and sells short the S&P500 index.2 The average residual return declines from 136 to 96 basis points per month. The long side of the SRR strategy, taken alone, provides most of the abnormal return.
l4 w E 5 p costs from exploiting it.
v)
Ln
MULTICOLLINEARITY OF MULTIPLE STRATEGIES
Multicollinearity of the strategy variables is an- other potential problem in studies of factors in market returns. When a variable is used in raw form to con- struct a strategy, without any attempt to immunize the strategy against other factors, the strategy weights are directly related to that variable. The mode of anal- ysis corresponds to a simple regression on that var- iable, and we can define the results as a “simple factor” of return. When that approach is taken, the major potential criticism of our study is that that var- iable may have served as a surrogate for other vari- ables more closely related to the subsequent abnormal returns.
In the present case, we have made each strat- egy orthogonal to the other strategy, to 55 industry groupings, and to 11 other “risk indexes,” which are continuous variables characterizing the stocks. This
1. Footnotes appear at the end of the article.
approach is subject to the criticism that this ortho- gonalization of the strategy weights may create wildly variable weightings because of muIticollineairity of these strategy variables with the other dimen sions.
Fortunately, this is not a problem. We delib- erately constructed the risk indexes so that multicol- linearity would not be severe. As a matter of fact, the time-series standard deviation of the B/P strategy re- turn discussed here is only 76 basis points, whereas the time-series variation of the simple B/P strategy return is 139 basis points. Both strategies have the same standardized exposure to the B/P ratio, so a reduction in the time-series variability can occur only if the risk reduction from immunizing the effects of other common factors has exceeded the risk increase due to higher specific variance from the wider varidble weightings. In other words, the multiple-factor strat- egy has substantially lower time-series risk, which confirms the benefits from orthogonalizing the weights.
Another important question related to the two tests is the extent to which they are independent of each other. Since the weightings are orthogonal a priori, we should expect the strategies to show in- dependent returns. The realized outcome was con- sistent with this: The correlation between the monthly residual returns on the EVP and SRR strategies was -.19 for the 45 overlapping months, which was insignificantly different from zero. A ”super strategy” that exploited a portfolio of the two strategies would therefore have achieved an even higher t-statistic than either strategy separately.
The B/P and SRR strategies are independent in another important sense. The B/P strategy corre- sponds to a ”slow idea,” and the SRR strategy to a ”fast idea.” Specifically, the B/P strategy exploits a decision criterion having data that are one to four months out of date (depending upon the month in the calendar quarter), and stocks purchased based on that criterion tend to be held for more than a year, on average. The SRR strategy exploits timely data, with 50% of the stocks in the portfolio traded at the end of the month. The success of two such diverse strategies tends to confirm, in our minds, the exist- ence of underlying pricing errors in the market, which can be imperfectly detected by either alternative in- strument.
POSSIBLE BIAS
O.ne potential problem in the study is a positive bias in the results due to errors in the recorded prices. The B/P and SRR strategies use instrumental variables for pricing error, and these will single out underval- ued securities, whether the low price is a true market
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fore, we conclude that - for this universe of stocks during this time period - the actual market prices were inefficient. The universe of stocks consists of 1400 of the largest companies in the Computstat data base. The time period is from 1980 to 1984. The stocks are priced largely on the NYSE, and a few are priced on the ASE, other regional exchanges, or NASDAQ.
The success of two such diverse instrumental variables in detecting market inefficiency suggests that there are still larger potential profits to be made, provided that the security analyst can identify the valuation errors that correlate with these instruments.
’ Investors can sell short only on up-ticks. It follows that in a declining market, the sell side of the SRR strategy would be difficult to implement in a timely fashion.
* This strategy could be implemented by selling S&P500 fu- tures contracts.
In an earlier version of the paper presented at the American Finance Association meeting (December 1984), we included only those stocks with a valid price within the last week of the month. We have since verified that the results also apply when all stocks which trade at any time within the month are included, with investment return calculated through to the last price.
16 CJ
# *
v)
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Sanjoy Basu. ”Investment Performance of Common Stocks in Relation to Their Price-earnings Ratios: A Test of the Efficient Market Hypothesis.” Iournal of Finance, June 1977, pp. 663-682. Eugene Fama. “Efficient Capital Markets: A Review iif Theory and Empirical Work.” Journal of Finance, May 1970, pp. 383- 417.
Lawrence Fisher. “Some New Stock Market Indices.” Journal of Business, January 1966, pp. 202-207. Robert Litzenberger and K.rishna Ramaswamy. “The Effect of Personal Taxes and Dividends on Capital Asset Prices .” Journal of Financial Economics, June 1979, pp. 163-195. Kenneth Reid. ”Average Returns to Equity Characteristics.” A paper presented at the Be::keley Program in Finance Seminar on Recent Evidence Concerni,ug Securities Market Efficiency, March 1982.
~ . ”Factors in the Pricing of Common Equity.” Unpublished doctoral dissertation, Graduate School of Business, University of California, Berkeley, June 1982. Marc Reinganum. “Misspecification of Capital Asset Pricing: Empirical Anomalies Based on Earnings’ Yields and h!!arket Values.” ]ournal of Financial Economics, March 1981, pp. l9-46. Barr Rosenberg and Vinay Marathe. ”Common Factors in Se- curity Returns: Microeconomic Determinants and Macroecon- omic Correlates.” Proceedings of the Seminar on the Analysis of Security Prices, May 1976, pp. 61-115. Barr Rosenberg and Andrew Rudd. ”Factor-related and Specific Returns of Common Stocks: Serial Correlation and Market Inefficiency.” Iournul of Finance, May 1982, pp. 543-554. Barr Rosenberg, Kenneth Reid, and Ronald Lanstein. “Factor Portfolios and Studies of Reward to Equity Character:istics.” A paper presented at the Quantitative Discussion Group, May 1982. Michael Solt and Meir Statman. “A Stock Return Regularity Based on Tobin’s Q-Ratio.” Unpublished manuscript, Leavey School of Business, University of Santa Clara, Santa Clara, California, November 1984. Timothy Sullivan. “A Note on Market Power and Returns to Stockholders.” Review of Economics and Statistics, February 1977, pp. 108-113.
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